A novel high-order solver for simulation of incompressible flows using the flux reconstruction method and lattice Boltzmann flux solver

In this paper, a novel high-order solver using the flux reconstruction (FR) method and the lattice Boltzmann flux solver (LBFS) is proposed for accurately and efficiently simulating nearly incompressible flows. The existing LBFSs are all based on the finite volume (FV) scheme. The present work extends the application of the LBFS for the first time by combining the high-order FR scheme. Compared with the traditional high-order incompressible Navier-stokes (N-S) solvers, no particular techniques are needed for the current FR-LBFS to overcome the dif-ficulty of the pressure-velocity coupling since the present method is a weakly compressible model. Moreover, unlike the traditional N-S solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly in FR-LBFS. In the present method, the common inviscid and viscous fluxes at the cell interface of the FR scheme are evaluated simultaneously by the local reconstruction of lattice Boltzmann equation (LBE) solution from macroscopic flow variables at solution points. Thus the discretization of the second-order partial derivative term is avoided. Compared with the high -order off-lattice Boltzmann methods (OLBM), the FR-LBFS is more stable, efficient and low-storage. No special techniques are needed to remove the stiffness of the governing equations which existing in the discrete velocity Boltzmann equation (DVBE). Compared with the high-order FV-LBFS, the FR-LBFS is compact for parallel computing by avoiding wide stencils on meshes. In addition, the present scheme has lower dissipation and can be more flexible to increase accuracy order. Numerical validations of the proposed method are implemented by simulating (a) Taylor-Green vortex problem, (b) steady plane Poiseuille flow, (c) lid-driven cavity flow, (d) laminar boundary layer and (e) flow past a square cylinder.